Evidence for (Infinitely Diverse) Non-Convex Mirrors Tristan - PowerPoint PPT Presentation

Evidence for 
 (Infinitely Diverse) 
 Non-Convex Mirrors Tristan Hübsch @ Southeastern Regional Mathematical String Theory Meeting 
 V-Tech University, Blacksburg VA; 2017.10.07 Departments of Physics & Astronomy and Mathematics, Howard University, Washington DC Department of Physics, Faculty of Natural Sciences, Novi Sad University, Serbia Department of Physics, University of Central Florida, Orlando FL

… — a mindmap Avoid the poles of Laurent polynomials gCI (C)NLSM Prehistory 
 Geometry: 
 Diffeo-Data 
 Holo-Data 
 1980s AAGGL 2015 
 ☛ H * ( X , ℤ ) ☛ H * ( X ) BH 2016/06 
 ☛ Chern classes 
 ☛ H * ( X,T ) 
 GvG 2017 ☛ Chern numbers 
 ☛ H * ( X, End T ) 
 ☛ Yukawa κ [ ω A , ω B , ω C ] ☛ Yukawa κ [ ϕ a , ϕ b , ϕ c ] ☛ p 1 [ ω A ] Toric GLSM Semiclassical Data ☛ phases 
 Geometry: Analysis ☛ phase-boundaries Quantum Data Textbooks † … 
 ☛ W 1993 … ✌"# ☛ A-discriminants 
 BH 2016/11 
 ☛ MP 1995 ☛ B-discriminants 
 BH 2017/ 10? Today! ☛ Yukawas 
 ☛ Instantons, GW

Non-Convex Mirror-Models Prehistory s c i s The Big Picture y h P Laurent GLSModels Phases & Discriminants M s …and in the Mirror c i a s y t h h “It doesn’t matter what it’s called, P …if… it has substance.” 
 S.-T. Yau 3

Pre-History (Where are We Coming From?)

Pre-History Classical Constructions Complete Intersections } Ex.: ( x–x 1 ) 2 +( y–y 1 ) 2 +( z–z 1 ) 2 = R 12 
 ( x–x 2 ) 2 +( y–y 2 ) 2 +( z–z 2 ) 2 = R 22 Algebraic (constraint) equations …in a well-understood “ambient” ( A ) Work over complex numbers …& incl. “infinity” (e.g., ℂℙ n ’s) For hypersurfaces : X ={ p ( x ) = 0} ⊂ A Just like gauge 
 Functions: [ f ( x )] X = [ f ( x ) ≃ f ( x ) + 휆 · p ( x )] A transformations Di ff erentials: [d x ] X = [d x ≃ d x + 휆 ·d p ( x )] A Homogeneity: ℂℙ n = U ( n +1)/[ U (1) × U ( n )] …with tensors Di ff erential r -forms on ℂℙ n are all U ( n +1) -tensors 5

The Big Picture (What are We Doing?)

Big Picture Superstrings = Framework for Models Gauged Linear Sigma Model (GLSM) — on the world-sheet Several “matter” fields + several “gauge” fields A μ ≃ A μ + ( ∇ μ λ ) Several coordinate functions – equivalence relations “Kinetic” part ( ∥ [ ∂ + q X A ] X ∥ 2 ): KE + gauge-matter coupling “Potential” part ( W ( X ) ): PE (gauge-invariant), “ F -terms” “Gauge” part ( ∥∂∧ A ∥ 2 + τ ·( ∂∧ A ) ): “ D -terms” & “ F.-I. terms” World-sheet matter & gauge symmetries are both complex E.g.: ( x 1 , x 2 , x 3 ) ≃ ( λ q 1 x 1 , λ q 2 x 2 , λ q 3 x 3 ) , λ ∈ ℂ * : ℙ 2 ( q 1 : q 2 : q 3 ) …makes sense if the fixed-point set is excised (forbidden) from ( x 1 , x 2 , x 3 ) ∈ ℂ 3 …or considered as an alternate (separate) location. } Gauge symmetry “stratifies” the X -field-space ⇒ spacetime & | vacuum ⟩ determined by min[ W ( X )] : hypersurface 8

Big Picture Toric Geometry Consider S 2 ≃ ℙ 1 : ξ (+1) ξ + 1 = η − 1 (–1) η Need at least two 
 (complex) coordinates: Match (the exponents) near the equator: (+1) N = (–1) S Symmetry: ξ → λ +1 ξ and η → λ –1 η , with λ ∈ ℂ * = ( ℂ ∖ {0}) Explicitly: λ = e i ( α + i β ) = e – β ·e i α = (real) rescaling · phase-change usual gauge 
 “thickened” S 1 transformation 9

Big Picture Toric Geometry More complicated examples: S 2 ⨯ S 2 An entire 2 nd sphere at every point of 1 st Orthogonal ↔ linearly independent Top-dim cones ↔ coord. patches 2-dim (enveloping) polytope ↔ ( ℂ ) 2-dim. geometry More complicated yet: “twisted” product T wisted torus S 1 ⨳ S 1 ( S 1 “twists” about S 1 ) 
 ( ≃ crystal w/oblique lattice $ ). Now ⨯ℂ : Hirzebruch ( ℂ ) surface, F 1 . $ n o g y “Slanting” (0,–1) → (– m ,–1) the bottom 
 l o p vertex (& two cones) encodes the “twist” g n i … F m = m -twisted ℙ 1 -bundle over ℙ 1 . n n a p …and so on: 4 textbooks worth! s 10

Toric Geometry x 3 Polytope Encoding x 1 The polytope encodes the space x 2 …but also its symmetries: m =1 x 4 Assign each vertex a (Cox) coordinate Read o ff cancelling relations v x 1 + 1 ~ v x 2 + 0 ~ v x 3 + 0 ~ v x 4 = 0 1 ~ ( x 1 , x 2 , x 3 , x 4 ) ' ( λ 1 x 1 , λ 1 x 2 , λ 0 x 3 , λ 0 x 4 ) v x 1 + m ~ v x 2 + 1 ~ v x 3 + 1 ~ v x 4 = 0 0 ~ ( x 1 , x 2 , x 3 , x 4 ) ' ( λ 0 x 1 , λ m x 2 , λ 1 x 3 , λ 1 x 4 ) Defines two independent (gauge) symmetries a GLSM w/gauge-invariant Lagrangian and | ground state ⟩ where KE = 0 = PE & (quantum) Hilbert space on it 11

BH Laurent GLSModels (and their Toric Geometry) A Generalized Construction of 
 Calabi-Yau Models and Mirror Symmetry arXiv:1611.10300

BH Laurent GLSMs — Proof-of-Concept — & Non-Convex Mirrors arXiv:1611.10300 2-torus in the Hirzebruch surface F m : “ Anticanonical” (Calabi-Yau, Ricci-flat) hypersurface in F m Toric description (0 , 1) ˆ e 2 =(0 , 1) N � � ? N R � Σ F 3 F 3 � 2 � 1 ( � 1 , 0) (1 , 0) � ˆ e 1 =( � 1 , 0) e 1 =(1 , 0) ˆ spanning polytope � 3 � 4 � � F 3 non-convex 
 � 3ˆ e 1 � ˆ e 2 =( � 3 , � 1) ( � m, � 1) ( � 3 , � 1) ( � m, � 1) for m>2 (…also, non-Fano for m> 2) The star-triangulation of the spanning polytope 
 defines the fan of the underlying toric variety 13

BH Laurent GLSMs — Proof-of-Concept — & Non-Convex Mirrors arXiv:1611.10300 The Newton polytope (polar of spanning polytope): The “standard” 
 ★ )°:={ u : ⟨ u , v ⟩≥ –1, v ∈ Δ ★ } ( Δ polar polytope 
 ν 2 is non-integral The spanning polytope φ 2 φ 1 � � � ∆ ? The “standard” 
 F 3 ∆ ? σ 2 σ 1 F 3 ⊂ N R ν 3 polar of the 
 σ 4 polar is not 
 φ 3 σ 3 ν 1 the spanning 
 ν 4 φ 4 polytope that 
 & we started with Is no good 
 F 3 ) � ) � = Conv( ∆ ? for mirror 
 (( ∆ ? F 3 ) % symmetry 6 = ∆ ? F 3 � 2 � 3 , � 1 14

BH Laurent GLSMs — Proof-of-Concept — & Non-Convex Mirrors arXiv:1611.10300 The oriented Newton polytope (trans-polar of spanning polytope): Constr uction (trans-polar) “Normal fan” ( φ 4 ) � Decompose Δ ⭑ into 
 Dual cones ↦ 
 convex faces θ i ; inside opening 
 $ trans-polar Find the (standard) polar 
 vertex-cones ( θ i )° for each (convex) face (Re)assemble parts dually 
 ( ν 1 ) � ν 2 to ( θ i ∩ θ j )° = [( θ i )°, ( θ j )°] 
 ( ν 4 ) � φ 2 φ 1 with “neighbors” ν 3 ν 1 φ 3 φ 4 ( φ 1 ) � ( φ 2 ) � ν 4 ( ν 2 ) � ( ν 3 ) � Agrees with standard (if obscure?) constructions… ( φ 3 ) � 15

BH Laurent GLSMs — Proof-of-Concept — & Non-Convex Mirrors arXiv:1611.10300 x 2 1 x 5 The oriented Newton polytope: ( � 1 , 4) ( � 1 , 1+ m ) 3 M � � F 3 specifies allowed monomials x 2 1 x 4 3 x 4 ( � 1 , 3) % The so-defined 2-tori 
 are all singular @(0,0,1) � F 3 � x 2 1 x 3 3 x 2 ( � 1 , 2) 4 …as each monomial has 
 at least an x 1 factor, so 
 x 1 x 2 x 2 x 2 1 x 2 3 x 3 3 4 ( � 1 , 1) (0 , 1) f ( x ) = x 1 · g ( x ) ( � ? F 3 ) � The extension 
 x 2 1 x 3 x 4 x 1 x 2 x 3 x 4 (0 , 0) 4 ( � 1 , 0) corresponds to 
 Laurent monomials: (0 , � 1) (1 , � 1) x 1 x 2 x 2 x 2 1 x 5 ( � 1 , � 1) x 2 4 4 9 % 2 ( 1, � 1 ) 7! � ( 2 m , � 1) ! ( 2 3 , � 1) = x 4 $# make the 2-tori Δ -regular. x 2 (1 , 1 � m ) ! (1 , � 2) 2 ; ( 1, � 2 ) 7! x 3 16

BH Laurent GLSMs — Proof-of-Concept — & Non-Convex Mirrors arXiv:1611.10300 The oriented Newton polytope: 9 is star-triangulable → a toric variety 9 ’ i r o 3 2 di ff ers from its convex hull by “flip-folded” simplices t 9 9 t ’ ’ a v H n o a + k m Associating coordinates to corners: i l , l h a o T d k u u + s P SP : x 1 =(–1,0), x 2 =(1,0), x 3 =(0,1), x 4 =(–3,–1) n a + o M h i i s k NP : y 1 =(–1,4), y 2 =(–1,–1), y 3 =(1,–1), y 4 =(1,–2) r s ☛ “multi-fans” a n K a ; � 7! v o Expressing each as a monomial in the others: h K 4 � y 5 � y 5 4 � x 2 � x 2 NP: x 2 1 x 5 3 � x 2 1 x 5 SP: y 2 1 y 2 2 � y 2 3 y 2 2 2 1 2 vs. x 4 x 3 y 4 y 3 2 3 2 3 P 2 P 2 (1:1:3) [5] (3:2:5) [10] 2 0 5 0 2 2 0 0 6 7 6 7 2 0 0 5 0 0 2 2 BHK 6 7 6 7 6 7 6 7 � 1 � 1 0 2 0 5 0 0 4 5 4 5 � 1 � 1 0 2 0 0 5 0 Mirror Construction 
 arXiv:hep-th/9201014 17